Properties

Label 17.1.86152660780...6144.1
Degree $17$
Signature $[1, 8]$
Discriminant $2^{30}\cdot 173^{8}$
Root discriminant $38.41$
Ramified primes $2, 173$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group $\PSL(2,16)$ (as 17T6)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74)
 
gp: K = bnfinit(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-74, 382, -776, 668, 520, -1084, -764, 484, 1238, 714, 16, -36, -6, 54, 2, -4, -1, 1]);
 

Normalized defining polynomial

\( x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(861526607800060221948166144=2^{30}\cdot 173^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $38.41$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 173$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{518449026921810234250726519} a^{16} - \frac{2903744299660867300931161}{518449026921810234250726519} a^{15} - \frac{145515969622780446567758039}{518449026921810234250726519} a^{14} + \frac{87449486712529578111145685}{518449026921810234250726519} a^{13} + \frac{90388261689503535099965497}{518449026921810234250726519} a^{12} + \frac{213755840339945972595489008}{518449026921810234250726519} a^{11} + \frac{72953974012171598161596854}{518449026921810234250726519} a^{10} + \frac{3137750212693082994430496}{518449026921810234250726519} a^{9} + \frac{256458172663421001979231745}{518449026921810234250726519} a^{8} - \frac{49611598799627003593660391}{518449026921810234250726519} a^{7} - \frac{27387611796406411645447552}{518449026921810234250726519} a^{6} + \frac{222961449395058127219738606}{518449026921810234250726519} a^{5} - \frac{63208606884273117780803614}{518449026921810234250726519} a^{4} + \frac{148709660332780900942449817}{518449026921810234250726519} a^{3} + \frac{30151660364221902163647120}{518449026921810234250726519} a^{2} + \frac{225255761602774742093792238}{518449026921810234250726519} a + \frac{1081393905446205624367037}{518449026921810234250726519}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{15}$, which has order $15$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 781146.759847 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$\PSL(2,16)$ (as 17T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 4080
The 17 conjugacy class representatives for $\PSL(2,16)$
Character table for $\PSL(2,16)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $17$ $17$ $17$ $17$ $15{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $17$ $17$ $15{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $17$ $17$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$173$$\Q_{173}$$x + 2$$1$$1$$0$Trivial$[\ ]$
173.4.2.1$x^{4} + 1557 x^{2} + 748225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
173.4.2.1$x^{4} + 1557 x^{2} + 748225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
173.4.2.1$x^{4} + 1557 x^{2} + 748225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
173.4.2.1$x^{4} + 1557 x^{2} + 748225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
15.861526607800060221948166144.240.a.a$15$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861526607800060221948166144.240.a.b$15$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861526607800060221948166144.240.a.c$15$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861526607800060221948166144.240.a.d$15$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861526607800060221948166144.240.a.e$15$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861526607800060221948166144.240.a.f$15$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861526607800060221948166144.240.a.g$15$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861526607800060221948166144.240.a.h$15$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
* 16.861526607800060221948166144.17t6.a.a$16$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $0$
17.861526607800060221948166144.51.a.a$17$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861526607800060221948166144.68.a.a$17$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861526607800060221948166144.68.a.b$17$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861526607800060221948166144.120.a.a$17$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861526607800060221948166144.120.a.b$17$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861526607800060221948166144.120.a.c$17$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861526607800060221948166144.120.a.d$17$ $ 2^{30} \cdot 173^{8}$ $x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + 382 x - 74$ $\PSL(2,16)$ (as 17T6) $1$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.